A collaborative group of researchers from Lawrence Livermore National Laboratory and the ESRF have been able to pin down the high-pressure lattice dynamics of the transition metal molybdenum by mapping its phonon energies under extremely high pressure. Using the inelastic X-ray scattering beamline ID28 at the European Synchrotron Radiation Facility (ESRF) and theoretical calculations, the team tracked the pressure evolution of a dynamical anomaly within molybdenum that has challenged scientists for over 40 years.
Much of the interest in the H-point phonon is derived from its anomalous increase in energy with increasing temperature. This observation stimulated numerous theoretical atte mpts to explain this strange behaviour. Changing the temperature or pressure is helpful in that it allows one to probe systems in different thermodynamic configurations. Indeed, the study of mater ials at high pressure is very useful for gaining insight into the nature of the chemical bonds in materials. Notably, the study of lattice dynamics at high pressures in general cannot be performed with neutrons due to the requirement of relatively large samples.
The group developed a new technique for preparing extremely small single Mo crystals of high crystalline quality [1]. These samples (40 micrometres in diameter by 20 micrometres thick) were placed into diamond anvil cells and taken to pressures as high as 40 GPa (400,000 atmospheres) to observe the evolution of the anomaly.
Fig. 1: A small molybednum single crystal loaded in the helium pressure medium. The photomicrograph was taken of the sample in situ at high pressure in the diamond anvil cell.
The researchers observed strong changes in the phonon dispersions at high pressure [2]. The most significant was a large difference in the Gruneisen parameter of modes at the H-point and those around q=0.65 along [001]. These differences lead to a dramatic decrease in the magnitude of the H-point anomaly with increasing pressure. Using theoretical codes developed to model molybdenum, the group showed that there is strong sensitivity of the H-point phonon on the electronic band structure. In fact, the decrease in the H-point anomaly required significant pressure induced broadening to match the experimental data. This implied a strong coupling between electronic states and phonons. With compression, the combination of an increase in the Fermi energy together with a broadening of the electronic states, leads to a significant decrease in this electron-phonon coupling. Thus, molybdenum becomes a much more 'normal' bcc metal at high pressures possibly explaining it's extraordinary stability in the bcc structure to pressures in excess of 400 GPa.
Fig. 2: Phonon dispersions in molybdenum at high pressure. The filled symbols show IXS data taken at 17 GPa at ID28, the open symbols are inelastic neutron scattering results at one atmosphere. Circles are longitudinal acoustic modes; squares transverse acoustic modes. Along [0] the triangles and squares show the two non-degenerate transverse acoustic modes TA[110]<-110> and TA[110]<001> respectively. The dashed lines show the calculations performed at one atmosphere, and the solid lines the calculations at 17 GPa.
Lattice dynamics of rubidium-IV, an incommensurate host-guest system
In recent years, a number of surprisingly complex crystal structures have been discovered in the elements at high pressures, in particular incommensurately modulated structures and incommensurate host-guest composite structures (see [1] for a review). The crystal structure of the high-pressure phase rubidium-IV shown in Figure 11 belongs to the group of incommensurate host-guest structures that have also been observed in the elements Na, K, Ba, Sr, Sc, As, Sb, and Bi. The structure comprises a framework of rubidium host atoms with open channels that are occupied by linear chains of rubidium guest atoms, and the periodicities of the host and guest subsystems are incommensurate with each other (i.e., they have a non-rational ratio). Although considerable progress has been made in determining the detailed crystal structures of the complex metallic phases at high pressure, little is known about their other physical properties, and the mechanisms that lead to their formation and stability are not yet fully understood.
Fig. 11: Inelastic X-ray scattering spectrum of Rb-IV at 17.0 GPa, with the scattering vector q = (0 0 3.2)h referring to the host lattice. The inset shows the composite crystal structure of Rb-IV with the rubidium host and guest atoms in blue and red, respectively.
We investigated the lattice dynamics in incommensurate composite Rb-IV by inelastic X-ray scattering (IXS) on beamline ID28. The focus was on the longitudinal-acoustic (LA) phonons along the direction of the incommensurate wavevector (parallel to the guest-atom chains). Calculations on simpler model systems predict these phonons to reflect the incommensurability most clearly. Phase IV of Rb is stable at pressures of 16 to 20 GPa at room temperature, and a high-quality single crystal of Rb-IV was grown in a diamond anvil high pressure cell. In the IXS experiment, the incident radiation was monochromatised at a photon energy of 17.8 keV, and two grazing-incidence mirrors focussed the X-rays onto the sample with a focal size of 25 x 60 µm. The spectrum of the scattered radiation was analysed by a high-resolution silicon crystal analyser to yield an overall energy resolution of 3 meV.
Figure 11 shows a typical IXS spectrum of Rb-IV along with its decomposition into the elastic line, the phonon excitation peaks and a constant background, which were obtained by least-squares fitting using the FIT28 software. From a series of IXS spectra collected for different momentum transfers Q, phonon dispersion curves were obtained as shown in Figure 12a. A central result of this study is the observation of two well-defined longitudinal-acoustic (LA)-type phonon branches along the chain direction. They are attributed to separate LA excitations in the host and the guest sublattices, which is a unique feature of an incommensurate composite crystal.
A series of dispersion curves was measured at different pressures, and from this the sound velocities of the host and guest excitations and their pressure dependences were determined (Figure 12b). While the absolute values of the sound velocities in the host and the guest are rather similar, their pressure dependences differ notably. A simple ball-and-spring model of Rb-IV with only one spring constant reproduces these observations semi-quantitatively. This suggests that the difference in the pressure dependences is determined largely by geometrical factors, i.e., by the spatial arrangement of the atoms rather than differences in the chemical bonding in the two subsystems.
There is only very weak coupling between the incommensurate host and the guest in Rb-IV, which raises a rather interesting question. Can the 1D chains of guest atoms in Rb-IV be considered a manifestation of the "monatomic linear chain" treated in solid-state physics textbooks to introduce the concepts of crystal lattice dynamics? The pressure dependence of the interatomic spacing in the guest-atom chains was measured in earlier structural studies and enables the spring constant in the linear chain model to be determined, and also its pressure dependence. On this basis, the sound velocity in the linear chains and its pressure dependence were modelled as shown in Figure 12b. The results are in excellent agreement with the IXS data for the guest-atom chains in the composite Rb-IV structure, which can thus be regarded as a manifestation of the monatomic linear chain model with regard to the LA phonons.
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